The linear charged black holes in AdS spacetime [1]} within a second-order phase transition show a scaling symmetry: at the critical point the state parameters scale respects to charge q, i.e., \(S\sim q^2,~P\sim q^{-2},~T\sim q^{-1}\)  [2]}. It is naturally to gauss whether there exists the scaling symmetry in the non-linear charged AdS black holes. As a generalization of the charged AdS Einstein-Maxwell black holes, it is interesting to explore new non-linear charged systems. Due to infinite self-energy of point like charges in Maxwell's theory [3]}, [4]}, [5]}, [6]}, [7]}, Born and Infeld proposed a generalization when the field is strong, bringing in non-linearities [8]}, [9]}. An interesting non-linear generalization of charged black holes involves a Yang-Mill field exponentially coupled to Einstein gravity, which possesses the conformal invariance and is easy to construct the analogues of the four-dimensional Reissner-Nordström black hole solutions in higher dimensions. Additionally several features of the Einstein-power-Yang-Mills (EPYM) gravity in extended thermodynamics have recently been studied [10]}, [11]}, [12]}.
